JEE PYQ: Units & Measurements - Question ID 770ee70afd35 (JEE Main 2023)

Core Formula & Concept:

In dimensional analysis, every physical quantity is expressed in terms of the fundamental dimensions: mass ($M$), length ($L$), time ($T$), electric current ($A$), etc. The key formulas and concepts used here are:

• Planck’s constant ($h$): From the relation $E = h \nu$, where $E$ is energy and $\nu$ is frequency. Energy has dimensions $[M^1 L^2 T^{-2}]$, and frequency has dimensions $[T^{-1}]$. Thus, $h = \frac{E}{\nu}$ has dimensions $[M^1 L^2 T^{-1}]$.
• Stopping potential ($V_s$): In the photoelectric effect, the maximum kinetic energy of emitted electrons is $eV_s$, where $e$ is the electronic charge. Since energy has dimensions $[M^1 L^2 T^{-2}]$, and charge $e$ has dimensions $[A^1 T^1]$, the stopping potential $V_s = \frac{E}{e}$ has dimensions $[M^1 L^2 T^{-3} A^{-1}]$.
• Work function ($\phi$): The work function is the minimum energy required to remove an electron from a metal surface. Since it is an energy, its dimensions are $[M^1 L^2 T^{-2}]$.
• Momentum ($p$): Momentum is defined as $p = mv$, where $m$ is mass and $v$ is velocity. Mass has dimensions $[M^1]$, and velocity has dimensions $[L^1 T^{-1}]$. Thus, momentum has dimensions $[M^1 L^1 T^{-1}]$.

Step-by-Step Derivation:

We will derive the dimensions of each quantity in List-I and match them with the corresponding entries in List-II.

1. Planck’s constant ($h$):

   From $E = h \nu$, we have: $h = \frac{E}{\nu}$ Dimensions of energy, $E = [M^1 L^2 T^{-2}]$.
   Dimensions of frequency, $\nu = [T^{-1}]$.
   Thus, $[h] = \frac{[M^1 L^2 T^{-2}]}{[T^{-1}]} = [M^1 L^2 T^{-1}]$ This matches III in List-II.

2. Stopping potential ($V_s$):

   The stopping potential is related to the maximum kinetic energy of photoelectrons by: $eV_s = K_{\text{max}}$ Dimensions of energy, $K_{\text{max}} = [M^1 L^2 T^{-2}]$.
   Dimensions of charge, $e = [A^1 T^1]$.
   Thus, $[V_s] = \frac{[M^1 L^2 T^{-2}]}{[A^1 T^1]} = [M^1 L^2 T^{-3} A^{-1}]$ This matches IV in List-II.

3. Work function ($\phi$):

   The work function is an energy, so its dimensions are: $[\phi] = [M^1 L^2 T^{-2}]$ This matches I in List-II.

4. Momentum ($p$):

   Momentum is given by $p = mv$.
   Dimensions of mass, $m = [M^1]$.
   Dimensions of velocity, $v = [L^1 T^{-1}]$.
   Thus, $[p] = [M^1] \cdot [L^1 T^{-1}] = [M^1 L^1 T^{-1}]$ This matches II in List-II.

Thus, the correct matching is:

• A → III
• B → IV
• C → I
• D → II

This corresponds to Option B.

Common Traps & Exam Tip:

Students often make the following mistakes in such questions:

• Confusing energy and power: Some students mistakenly assign the dimensions of power ($[M^1 L^2 T^{-3}]$) to energy or work function. Remember that power is energy per unit time, so their dimensions differ by a factor of $T^{-1}$.
• Incorrect dimensions for stopping potential: Stopping potential is often confused with electric potential ($V$), which has dimensions $[M^1 L^2 T^{-3} A^{-1}]$. However, students may incorrectly assign it the dimensions of energy or voltage without considering the role of charge.
• Planck’s constant dimensions: Some students recall that $h$ is related to angular momentum and mistakenly assign it the dimensions of angular momentum ($[M^1 L^2 T^{-1}]$), which is correct, but they may confuse it with other quantities like torque or energy.
• Momentum vs. force: Momentum ($[M^1 L^1 T^{-1}]$) and force ($[M^1 L^1 T^{-2}]$) differ by a factor of $T^{-1}$. Students sometimes mix these up, especially under exam pressure.

Exam Tip: Always derive the dimensions from first principles rather than relying on memory. For example, start with the defining equation (e.g., $E = h \nu$ for Planck’s constant) and substitute the known dimensions of other quantities to find the unknown dimensions. This approach minimizes errors.